The zeroes of the quadratic polynomial x2 + kx + k, k $\neq$ 0 ______.

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The zeroes of the quadratic polynomial x2 + kx + k, k $\neq$ 0 cannot both be positive. Explanation: Let p(x) = x2 + kx + k, k $\neq$ 0 On comparing p(x) with ax2 + bx + c, we get Now, a = 1, b = k and c = k `x = (-b +- sqrt(b^2 - 4ac))/(2a)` ....[By quadratic formula] = `(-k +- sqrt(k^2 - 4k))/(2 xx 1) = (-k +- sqrt(k(k - 4)))/2, k $\neq$ 0` Here, we see that k(k − 4) > 0 $\Rightarrow$ k ∈ (-oo, 0) u (4, oo) Now, we know that In quadratic polynomial ax2 + bx + c If a > 0 b > 0 c > 0 or a < 0 b < 0 c < 0 Then the polynomial has always all negative zeroes. And if a > 0, c < 0 or a < 0, c > 0 Then the polynomial has always zeroes of opposite sign Case I: If k ∈ (-oo, 0) i.e., k < 0 $\Rightarrow$ a = 1 > 0, b, c = k < 0 So, both zeroes are of opposite sign. Case II: If k ∈ (4, oo) i.e., k $\geq$ 4 $\Rightarrow$ a = 1 > 0, b, c > 4 So, both zeroes are negative. Hence, in any case zeroes of the given quadratic polynomial cannot both be positive. [Solution diagram: 269277_solution_1.png]

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